I am between 15 and 16 years of age, currently training in Shūkōkai karate, and am close to achieving my 1st Dan black belt. I was debating whether to try another style alongside my current training. What would you recommend?
My opinion would be for you to attend some trial classes and have a feel of what suits you. Some people would like to try something that has a familiar feel them, in your case, it would be something like kick boxing or etc. Others would like to try something that is totally foreign to them, such as Wrestling or BJJ.
Whatever you choose, do try to 'unlearn' whatever you know from you previous art. Try to be like a newborn, absorbing whatever things they teach, even if it contradicts to what you know. As you get better and more experience, slowly integrate what you have learned in Karate and see what works and what don't.
Coming from a mainly 'stand-up' background, I was like a fish out of water doing BJJ. Things like 'center of gravity' or 'stances' doesn't have much role in it. Instead things like 'leverage' have a more significant impact.
How far from the tree do you want to fall off?
Are you looking to go way deeper into the way you already took? Then I would suggest you to try a different Karate style that feeds or is feeded by Shukokai. Goju-Ryu for example, or a different approach like Shotokan.
If you are looking to be a more complete martial artists, I guess the sensitive way would be to go either towards aikido or jiu-jitsu for throws and floor fighting techniques, towards Kyokushin karate for hardening and full contact training, or towards some styles of wushu (kung fu) like Wing Chun for speed and short-range swift attacks.
Last but not least, another suggestion might be to take an advanced technique or group of techniques of your style, or from a relative style of your own, and specialize yourself in such techniques towards perfection.
Just remember, sometimes the fruit must fall off the tree before it can serve its own purpose.